Schur-positivity conjecture for the decomposition poset

The conjecture asserts that moving upward in the decomposition poset increases the corresponding product in Schur order. Although Loktev, Papi, and Ponomarev are associated with a row-shuffle result, the retrieved literature does not identify that theorem with the full conjecture; a 2012 source explicitly states the general case as conjectural.

Known results

• Fomin, Fulton, Li, and Poon (2004) proved several significant cases of the related type-$A$ positive-difference conjecture.
• Lam, Postnikov, and Pylyavskyy (2005) proved the partition-sorting formulation of that conjecture.
• Loktev, Papi, and Ponomarev (2007) proved the row-shuffle statement in type-$A$.
• For the decomposition-poset formulation, tensor-product inclusions are known when the weight is a multiple of a fundamental minuscule weight, and for type-$A_2$ with two factors (2012).

Current status (as of September 2026): The full decomposition-poset conjecture remains open in the retrieved evidence; related type-$A$ results and the recorded special cases are settled.